Intensive studies of hyperbolic metamaterials (HMMs) are induced by unique optical properties of this type of artificial media associated with their hyperbolic dispersion. Special attention is attracted to the nonlinear optical response of HMMs, which reveals anomalous behavior in definite spectral regions. Third-order nonlinear optical self-action effects that are perspective for applications were analyzed numerically, whereas such experiments have not been performed up to now. In this work we study experimentally the effects of the nonlinear absorption and refraction in ordered arrays of gold nanorods in porous aluminum oxide. We demonstrate strong enhancement and sign reversal of these effects in the vicinity of the epsilon-near-zero spectral point due to the resonant light localization and transition from elliptical to hyperbolic dispersion regimes.